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Swedish Mathematical Competition
2023 Swedish Mathematical Competition
6
6
Part of
2023 Swedish Mathematical Competition
Problems
(1)
every rational in (0,1) i sum of 1/k(k+1)
Source: 2023 Swedish Mathematical Competition p6
3/24/2024
Prove that every rational number
x
x
x
in the interval
(
0
,
1
)
(0, 1)
(
0
,
1
)
can be written as a finite sum of different fractions of the type
1
k
(
k
+
1
)
\frac{1}{k(k + 1)}
k
(
k
+
1
)
1
, that is, different elements in the sequence
1
2
\frac12
2
1
,
1
6
\frac{1}{6}
6
1
,
1
12
\frac{1}{12}
12
1
,
.
.
.
...
...
.
algebra
number theory